EEE 321 Signals and Systems Ankara University Faculty of Engineering Electrical and Electronics Engineering Department
EEE 321Signals and Systems
Ankara University
Faculty of Engineering
Electrical and Electronics Engineering Department
Power andEnergy SignalsEEE321 Signals and Systems
Lecture 4
Ankara UniversityElectrical and Electronics Engineering Department, EEE321
Signals and Systems
Agenda
• Power
• Energy
• Harmonically related complex exponentials
Ankara UniversityElectrical and Electronics Engineering Department, EEE321
Signals and Systems
• Periodic signals are examples of signals with infinite total energy, but finite average power.
• Consider periodic complex exponential signal 𝑒𝑗𝜔𝑜𝑡.
Total energy and average power of the signal over one period are
𝐸𝑇𝑜 =
0
𝑇𝑜
|𝑒𝑗𝜔𝑜𝑡|2𝑑𝑡
=
0
𝑇𝑜
1𝑑𝑡 = 𝑇𝑜
𝑃𝑇𝑜 =1
𝑇𝑜𝐸𝑇𝑜 = 1
It is also clear that 𝐸∞ = ∞
and
𝑃∞ = lim𝑇→∞
1
2𝑇
−𝑇
𝑇
|𝑒𝑗𝜔𝑜𝑡|2𝑑𝑡 = 1
Power and Energy of Periodic Signals
• Relationship between the fundamental frequency and period for continuous-time signals
ω1>ω2>ω3 T1<T2<T3
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Sets of harmonically related complex exponentials
𝜙𝑘 𝑡 = 𝑒𝑗𝑘𝜔𝑜𝑡, 𝑘 = 0,±1,±2,…
Since 𝑒𝑗𝜔𝑇𝑜 = 1 implies that 𝜔𝑇𝑜 = 2𝜋𝑘, 𝑘 = 0,±1,±2,… (𝜔𝑇𝑜 is multiple of 𝜋𝑘)
𝜔𝑜 =2𝜋
𝑇𝑜For k=0, 𝜙𝑘 𝑡 is a constant, while for other values of k, 𝜙𝑘 𝑡 is periodic withfundamental frequency |k|ωo and fundamental period
2𝜋
|𝑘|𝜔𝑜=𝑇𝑜|𝑘|
The kth harmonic 𝜙𝑘 𝑡 is still periodic with period To as well, as it goes throughexactly |k| of its fundamental periods during any time interval of length To.
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Example *
Plotting the magnitude of the signal 𝑥 𝑡 = 𝑒𝑗2𝑡 + 𝑒𝑗3𝑡
It can be rewritten as𝑥 𝑡 = 𝑒𝑗2.5𝑡(𝑒−𝑗0.5𝑡 + 𝑒𝑗0.5𝑡)
Using Euler’s relation𝑥 𝑡 = 2𝑒𝑗2.5𝑡 cos 0.5𝑡
Therefore|𝑥 𝑡 | = 2|cos 0.5𝑡 |
Note that |𝑒𝑗2.5𝑡|=1.
* Example 1.5. Signals and Systems, A.V. Oppenheim, A. S. Willsky with S. H. NawabAnkara University
Electrical and Electronics Engineering Department, EEE321 Signals and Systems
Example (cont.)*
* Example 1.5. Signals and Systems, A.V. Oppenheim, A. S. Willsky with S. H. Nawab
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Signals and Systems
General Complex Exponential Signals
𝑥 𝑡 = 𝐶𝑒𝑎𝑡
If C and a are expressed in polar and rectengular form, respectively
𝐶 = |𝐶|𝑒𝑗𝜃
𝑎 = 𝑟 + 𝑗𝜔𝑜
Then 𝐶𝑒𝑎𝑡 = 𝐶 𝑒𝑗𝜃𝑒 𝑟+𝑗𝜔𝑜 𝑡 = |𝐶|𝑒𝑟𝑡𝑒𝑗(𝜔𝑜𝑡+𝜃)
By using Euler’s equation, 𝐶𝑒𝑎𝑡 = 𝐶 𝑒𝑟𝑡 cos 𝜔𝑜𝑡 + 𝜃 + 𝑗 𝐶 𝑒𝑟𝑡 sin 𝜔𝑜𝑡 + 𝜃
Therefore, for r=0, the real and imaginary parts of a complex exponential aresinusoidal. For r>0, the signal is sinusoidal multiplied by growing exponential. If r<0, the signal is sinusoidal multiplied by decaying exponential.
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Discrete-Time Complex Exponential and Sinusoidal Signals
Discrete-Time Complex Exponential Signals
𝑥 𝑛 = 𝐶𝛼𝑛
where C and α are complex numbers in general.
If α=𝑒𝛽 , then 𝑥 𝑛 = 𝐶𝑒𝛽𝑛
Real-exponential signals: If C and α are real x[n] is called real exponential signal.
* If α is 1, then x[n] is constant, if α is than x[n] alternates between –C and +C.
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• Discrete-Time Sinusoidal Signals:
If β is purely imaginary than |α|=1. Specifically, consider𝑥 𝑛 = 𝑒𝑗𝜔𝑜𝑛
which has infitine total energy, but finite average power.
As in continuous-time case this is related to𝑥 𝑛 = 𝐴cos 𝜔𝑜𝑛 + 𝜑
which also has infinite total energy, but finite average power.
Since n is dimensionless, both 𝜔𝑜 and 𝜑 will have units of radians.
Also Euler’s equation is 𝑒𝑗𝜔𝑜𝑛 = cos 𝜔𝑜𝑛 + 𝑗𝑠𝑖𝑛 𝜔𝑜𝑛
and therefore
𝐴cos 𝜔𝑜𝑛 + 𝜑 =𝐴
2𝑒𝑗𝜑𝑒𝑗𝜔𝑜𝑛 +
𝐴
2𝑒−𝑗𝜑𝑒−𝑗𝜔𝑜𝑛
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